The finite 
-adic
complexity of a string of digits
in base 
measures the smallest rational
number height needed to extend that string as a 
-adic
expansion. For integers 
and 
, represent the string by
 |
(1)
|
where 
and 
is the least significant digit. Define
 |
(2)
|
where 
and 
are integers. The finite 
-adic complexity is
 |
(3)
|
The condition on the greatest common divisor makes 
invertible modulo 
, and the congruence fixes
the first 
digits. The logarithm is unrounded.
For 
,
this is finite 2-adic complexity. For
composite bases, the definition still uses congruence
modulo 
and does not assume that the base is a prime number.
For uniformly chosen strings, the expectation value is
 |
(4)
|
Babanskyy (2026) reported the uniform bound
 |
(5)
|
for every integer 
and 
. The same work claims 
when 
is a prime power and 
when 
has exactly two distinct prime
factors. In contrast, 
, so the bound 
does not hold in every base. The
optimal additive constant remains unknown.
The author reports OpenAI GPT assistance with the mathematics and software. As of Oct. 15, 2026, independent specialist review of the written reductions and infinite estimates, and independent reproduction of the complete computational certificates, had not been reported. No Lean formalization was supplied. These statements concern the unrounded rational complexity, not a rounded register length (Babanskyy 2026).
